UltraIterator: Iteration Skipping

Fellow fractal explorer Sam Monnier has recently done great work with some interesting new space-filling fractal formulae, including a new (to me) method for coloring densely detailed fractals. The idea is to skip iterations in an otherwise normal coloring formula, which can really help to introduce variations in what details of the fractal shape are resolved. Skipping (or weighting) makes it possible to build a much more interesting structure to a composite, with isolated detail ranges contributing different aspects of the fractal shape to the final result.

In order to play with this I made modified versions of two of the coloring methods that I use a lot: Exponential Smoothing and Triangle Inequality Average. I added controls for both modulo-based iteration skipping and weighting of iterations using a cosine function. Together the two have been extremely useful and they make it possible to really change the way the color gradient is looked up by the form of the fractal.

The following is one of the most interesting things I've found using iteration skipping, it's a phoenixDoubleNova with 6 layers:

Its neighborhood, with wiggly brainy shapes:

Iteration skipping helped in these Nova fractals both to increase the chaos and make it look like something specific is happening:

These phoenixDoubleNova got some of their shape and dynamic range from skipping layers:

Pseudo-basin boundaries are sculpted by the skipping in interesting ways. The next few are from a particular Julia-lake (shown first) that overlaps the edge of a mandelfield. When these plateaux straddle the area between open lakes and closed, they apply pressure to the dynamics of the foam, preventing the pseudo-lakes from closing as fast (near the edge it takes more iterations for them to close):

Here iteration skipping gave rise to an altogether different look to the typical phoenixDoubleNova spiral network - isolating the julia foam atoms in a way that for some reason reminded me of teeth: